Pure Mathematics
Vol. 12  No. 04 ( 2022 ), Article ID: 50396 , 4 pages
10.12677/PM.2022.124062

一类Newman多项式的性质

李昌吉

阿坝师范学院,藏汉双语学院,四川 汶川

收稿日期:2022年3月10日;录用日期:2022年4月13日;发布日期:2022年4月20日

摘要

与系数相关的表达式的极值问题是Newman多项式相关研究中的一个热点。令 h i ( x ) 是一类系数全为1的Newman多项式,借助不等式和组合的方法,讨论了与 h i 3 ( x ) h i 4 ( x ) 系数相关表达式的取值,给出了该表达式的极值,从n的不同取值对结论进行了推广。

关键词

Newman多项式,系数,极值性质

Properties of a Class of Newman Polynomials

Changji Li

Tibetan-Chinese Bilingual School, Aba Teachers University, Wenchuan Sichuan

Received: Mar. 10th, 2022; accepted: Apr. 13th, 2022; published: Apr. 20th, 2022

ABSTRACT

The extreme value problem of the expression related to coefficients is a hot spot in the research of Newman polynomials. Letting h i ( x ) be a kind of Newman polynomials with all coefficients of 1, the value of the coefficient correlation expression of the h i 3 ( x ) and h i 4 ( x ) is discussed by method of inequality and combination,and the extremal properties of the expression are given, and the conclusion is generalized from different values of n.

Keywords:Newman Polynomials, Coefficients, Extremal Properties

Copyright © 2022 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言及结论

多项式是代数中的重要内容之一,系数受限的多项式及其相关性质是多项式研究中的热点问题之一。Newman多项式是指形如 f i ( x ) = i = 0 m a i x i a i { 0 , 1 } 的多项式,这是一类系数受限的特殊多项式。有关Newman多项式的研究成果较多,如文献 [1] - [8]。一些学者聚焦于研究Newman多项式中关于系数极值性质,取得了一定的成果。如文献 [9] 研究了Newman多项式导数的一些极值性质。记 ( # f i ) 为多项式 f i ( x ) 中系数非零的项数, ζ ( f i n ) 是多项式 f i n 展开式所有项中的最大系数, γ ( n ) = deg ( f i ) ζ ( f i n ) ( # f i ) n 。文献 [10] 给出当 ( # f i ) = o ( deg f i ) 时,可以推出 lim i inf ( γ ( 2 ) ) 1 。文献 [11] 指出当条件 ( # f i ) = o ( deg f i ) 取消后, lim i inf ( γ ( 2 ) ) 发生变化,并得出 lim i inf ( γ ( 2 ) ) = 8 9 ,并猜测此时有 inf ( γ ( 2 ) ) 8 9 。本文将研究在 ( # f i ) = o ( deg f i ) 情形下,一类特定形式的Newman多项式在 i 时的 inf ( γ ( 3 ) ) inf ( γ ( 4 ) ) 的极值问题。本文中研究的Newman多项式的类型是形如: h i ( x ) = i = 0 m a i x i a i = 1 的多项式,在此条件下有 ( # h i ) = o ( deg h i ) ,并得出结论如下:

定理1 当 h i ( x ) = i = 0 m a i x i a i = 1 时,有 lim i inf ( γ ( 3 ) ) = 3 4

定理2 当 h i ( x ) = i = 0 m a i x i a i = 1 时,有 lim i inf ( γ ( 4 ) ) = 2 3

2. 定理证明

2.1. 定理1的证明

h i ( x ) = i = 0 m a i x i = 1 + x + x 2 + x 3 + + x i 1 + x i

易知, ( # h i ) = i + 1 ( deg h i ) = i ,又

h i 3 ( x ) = ( 1 + x + + x i 1 + x i ) 3 = 1 + 3 x + 6 x 2 + + i ( i + 1 ) 2 x i 1 + ( i + 1 ) ( i + 2 ) 2 x i + i 2 + 5 i 2 x i + 1 + i 2 + 6 i 6 2 x i + 2 + + i 2 + 3 i + 2 2 x 2 i + i ( i + 1 ) 2 x 2 i + 1 + ( i 1 ) i 2 x 2 i + 2 + + 3 x 3 i 1 + x 3 i

i 0 ( mod 2 ) 时,多项式 h i 3 ( x ) x 3 2 i 的系数最大,此时有

ζ ( h i 3 ) = ( i 2 + 1 ) + ( i 2 + 2 ) + + ( i 1 ) + i + ( i 1 ) + ( i 2 + 2 ) + ( i 2 + 1 ) = 3 ( i + 1 ) ( i + 2 ) 4

所以 lim i inf ( γ ( 3 ) ) = lim i i 3 ( i + 1 ) ( i + 2 ) 4 ( i + 1 ) 3 = 3 4

i 1 ( mod 2 ) 时,多项式 h i 3 ( x ) x 3 i 1 2 x 3 i + 1 2 的系数最大,此时有

ζ ( h i 3 ) = i + 3 2 + i + 5 2 + + i + ( i + 1 ) + i + + i + 5 2 + i + 3 2 + i + 1 2 = 3 ( i + 1 ) 2 4

所以 lim i inf ( γ ( 3 ) ) = lim i i 3 ( i + 1 ) 2 4 ( i + 1 ) 3 = 3 4

综上,对任意正整数 i ,均有 lim i inf ( γ ( 3 ) ) = 3 4 ,定理1得证。

2.2. 定理2的证明

h i ( x ) = i = 0 m a i x i = 1 + x + x 2 + x 3 + + x i 1 + x i

易知, ( # h i ) = i + 1 ( deg h i ) = i ,又

h i 4 ( x ) = ( 1 + x + + x i 1 + x i ) 4 = ( 1 + 2 x + 3 x 2 + 4 x 3 + + i x i 1 + ( i + 1 ) x i + ( i 1 ) x i + 1 + + 3 x 2 i 2 + 2 x 2 i 1 + x 2 i ) 2 = r = 0 i ( j + k = r + 2 , j , k > 0 j k ) x r + ( i 1 + ( i + 1 ) 2 + i 3 + ( i 1 ) 4 + + 2 ( i + 1 ) + 1 i ) x i + 1 + ( ( i 1 ) 1 + i 2 + ( i + 1 ) 3 + i 4 + + 3 ( i + 1 ) + 2 i + 1 ( i 1 ) ) x i + 2

+ ( ( i 2 ) 1 + ( i 1 ) 2 + i 3 + ( i + 1 ) 4 + i 5 + + 2 ( i 1 ) + 1 ( i 2 ) ) x i + 3 + + ( 1 1 + 2 2 + + i i + ( i + 1 ) ( i + 1 ) + i i + + 2 2 + 1 1 ) x 2 i + ( 1 2 + 2 3 + + ( i 1 ) i + i ( i + 1 ) + ( i + 1 ) i + i ( i 1 ) + + 3 2 + 2 1 ) x 2 i + 1 + ( 1 3 + 2 4 + + ( i 1 ) ( i + 1 ) + i i + ( i + 1 ) ( i 1 ) + i ( i 2 ) + + 4 2 + 3 1 ) x 2 i + 2

+ + ( 1 ( i + 1 ) + 2 i + 3 ( i 1 ) + ( i 1 ) 3 + i 2 + ( i + 1 ) 1 ) x 3 i + ( j + k = i + 2 , j , k > 0 j k ) x 3 i + ( j + k = i + 1 , j , k > 0 j k ) x 3 i + 1 + ( j + k = i , j , k > 0 j k ) x 3 i + 2 + + ( j + k = 3 , j , k > 0 j k ) x 4 i 1 + ( j + k = 2 , j , k > 0 j k ) x 4 i

结合排序不等式,易知多项式 h i 4 ( x ) 展开式中 x 2 i 的系数最大,此时有

ζ ( h i 4 ) = 1 2 + 2 2 + + i 2 + ( i + 1 ) 2 + i 2 + + 2 2 + 1 2 = 2 i ( i + 1 ) ( 2 i + 1 ) 6 + ( i + 1 ) 2 = ( i + 1 ) ( 2 i 2 + 4 i + 3 ) 3

所以 lim i inf ( γ ( 4 ) ) = lim i i ( i + 1 ) ( 2 i 2 + 4 i + 3 ) 3 ( i + 1 ) 4 = 2 3

综上,定理2得证。

3. 研究展望

本文主要探讨了一类Newman多项式 f i 中关于相关系数的表达式 γ ( n ) = deg ( f i ) ζ ( f i n ) ( # f i ) n ( ( # f i ) = o ( deg f i ) )的极值问题,将 n 的值从2的情形推广到了3和4的情形。当条件 ( # f i ) = o ( deg f i ) 取消时,本文猜测 n = 3 和4时 γ ( n ) = deg ( f i ) ζ ( f i n ) ( # f i ) n 的极值情况将会和 n = 2 时发生改变的情形相似,也会发生改变,在此情形下, ( # f i ) deg ( f i ) 的极值相应会有怎样的变化,这些将作为下一步研究的方向。

基金项目

阿坝师范学院科研项目(20170101, ASB21-04, 202007013)。

文章引用

李昌吉. 一类Newman多项式的性质
Properties of a Class of Newman Polynomials[J]. 理论数学, 2022, 12(04): 561-564. https://doi.org/10.12677/PM.2022.124062

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